Nonlinear Equations - UFRJ

viii
FOREWORD
solving
( x
y x)
+ y (x + y) 2 = 325
216
( x
y x)
+ y + 2xy + (x − y) 2 = 91
36 .
It is believed to date from the end of the first dynasty of Babylon
(16 th century BC). Yet, very little is known on how to efficiently
solve nonlinear equations, and even counting the number of solutions
of a specific nonlinear equation can be extremely challenging.
These notes
These notes correspond to a short course during the 28 th Colóquio
Brasileiro de Matemática, held in Rio de Janeiro in July 2011. My
plan is to let them grow into a book that can be used for a graduate
course on the mathematics of nonlinear equation solving.
Several topics are not properly covered yet. Subjects such as
univariate solving, modern elimination theory, straight line programs,
random matrices, toric homotopy, finding start systems for homotopy,
how to certify degenerate roots or curves of solutions [83], tropical
geometry, Diophantine approximation, real solving and Khovanskii’s
theory of fewnomials [49] should certainly deserve extra chapters.
Other topics may be a moving subject (see below).
At this time, these notes are untested and unrefereed. I will keep
an errata in my page, http://www.labma.ufrj.br/~gregorio
Most of the material here is known, but some of it is new. To
my knowledge, the systematic study of spaces of complex fewnomial
spaces (nicknamed fewspaces in Definition 5.2) is not available in
other books (though Theorem 5.11 was well known).
The theory of condition numbers for sparse polynomial systems
(Chapter 8) presents clarifications over previous tentatives (to my
knowledge only [58] and [59]). Theorem 8.23 is a strong improvement
over known bounds.
Newton iteration and ‘alpha theory’ seem to be more mature topics,
where sharp constants are known. However, I am unaware of